Contractive Iterated Function Systems Enriched with Nonexpansive Maps

Motivated by a recent paper of Leśniak and Snigireva [Iterated function systems enriched with symmetry, preprint], we investigate the properties of the semiattractor AF∪G∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{\mathcal {F}\cup \mathcal {G}}^*$$\end{document} of an IFS F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}$$\end{document} enriched by some other IFS G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}$$\end{document}. We show that in natural cases, the semiattractor AF∪G∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{\mathcal {F}\cup \mathcal {G}}^*$$\end{document} is in fact the attractor of certain IFSs related naturally with the IFSs F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}$$\end{document} and G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}$$\end{document}. We also give an example when AF∪G∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{\mathcal {F}\cup \mathcal {G}}^*$$\end{document} is not compact, yet still being the attractor of considered related IFSs. Finally, we use presented machinery to prove that the so called lower transition attractors due to Vince are semiattractors of enriched IFSs.


Introduction
In the paper we study the following problem: Assume that F is an iterated function system (IFS for short) which has the attractor A F or the semiattractor A * F , and let G be any IFS on the same underlying space. What can we say about the semiattractor A * F∪G of the IFS F ∪ G? The motivation of our studies came from the recent paper of Leśniak and Snigireva [16], in which the authors consider the case when F is a finite family of Banach contractions and G is singleton consisting of periodic symmetry (that is, the isometry whose certain iteration is the identity) with a fixed point. Roughly speaking, the main results of [16] show that in considered cases, the semiattractor A * F∪G is in fact the attractor of certain IFSs which are somehow related with F and G. Moreover, the authors of [16] prove that the semiattractor A * F∪G can be recovered by the famous chaos game algorithm. In the present paper, we extend the framework. Instead of a single periodic isometry, in our main results we assume that G is just an IFS consisting of nonexpansive maps. We also weaken the contractive assumptions on maps from the IFS F and allow F to be infinite. Let us remark that dealing with the theory of infinite IFSs is necessary in considered problem-on the one hand, in the reasonings we come across to monoinds and groups generated by the IFS G which in most cases are infinite, and on the other hand, we give a relatively simple example in which the semiattractor A * G∪F is closed and bounded, yet not compact, so cannot be an attractor of a finite IFS.
The paper is organized as follows: In the next section we present basic definitions and facts, among them the notion of iterated function system, its attractor and semiattractor in the sense of Lasota and Myjak. We also present basic results on the existence of attractors and semiattractors.
Then, in Sect. 3, we present the first main result of the paper. Namely, using just set-algebraic reasonings, we present different descriptions of the semiattractor A * F∪G in a very general framework-we just assume that the semiattractor A * F of the IFS F exists. More precisely, we show that the semiattractor A * F∪G is the semiattractor (or some its modification) of certain IFSs, related somehow with F and G.
In Sect. 4 we prove that in natural case when F admits the attractor A F and G consists of nonexpansive maps, the semiattractor A * F∪G is in fact the attractor of IFSs considered in Sect. 3. Additional properties of A * F∪G are shown under stronger assumption that the monoid generated by G is a group. We also give some sufficient conditions under which the semiattractor is compact.
In Sect. 5 we present a relatively natural example of IFSs F and G which satisfy the assumptions considered in Sect. 4, but whose semiattractor A * F∪G is not compact. In fact, the IFS F consists of just one constant map, and G consists of the so-called rotative isometry (which, in a sense, can be considered as "almost periodic isometry"). This example shows limitations in searching for compact semiattractors of enriching IFSs.
Then, in Sect. 6 we present some natural examples of enriched IFSs on the plane and their attractors. We focus on the ones which cannot be handled by the framework from the paper [16], that is, we will enrich IFSs F by maps which are not periodic isometries.
Finally, in Sect. 7 we use presented machinery to prove that lower transition attractors considered recently by Vince [21] are attractors of enriched IFSs.
At the end of the introductory part, lest us remark that the authors of [16] were mainly interested in the case when the obtained semiattractor A * F∪G is symmetric w.r.t. the maps from G, i.e., when g(A * F∪G ) = A * F∪G for Definition 2.1. By an iterated function system (IFS for short) we will mean any family F of continuous selfmaps of a metric space X. If F is an IFS, then by F F we will denote the Hutchinson operator adjusted to F, that is, the map F F : 2 X → 2 X defined by: We will say that an IFS F is -bounded, if for every bounded set D, the set F F (D) is also bounded; -compact, if for every compact set D, the set F F (D) is also compact.
In other words, an IFS F is bounded iff the restriction

Remark 2.2.
(1) Each finite IFS F = {f i : i ∈ I} is compact and for every K ∈ K(X), we have that F F (K) = i∈I f i (K) (that is, we may omit the closure). (2) Each finite IFS consisting of Lipschitz maps is bounded.
(3) Let F be an IFS so that sup{Lip(f ) : f ∈ F} < ∞. Then the following conditions are equivalent: Items (1) and (2) are obvious (in fact, (2) follows from (4)) and (4) follows from (3). We just make a small comment on (3). The implication (i) → (ii) is trival. To see the opposite one, take any nonempty bounded set D ⊂ X and let L := sup{d(x, z 0 ) : x ∈ D}, M := diam({f (z 0 ) : f ∈ F}) and N := sup{Lip(f ) : f ∈ F}. For every x, y ∈ D and f, g ∈ F, we have Remark 2.3. Most authors assume that IFSs are finite families of maps and that the Hutchinson operators are defined on the hyperspaces K(X) or (rarely) on C b (X). As mentioned in the Introduction, our setting naturally moves the discussion to infinite IFSs. Such IFSs were considered in the literature, for example in [8,9,12,17] or [20]. In fact, item (3) of the above Remark for nonexpansive maps was noted in [12] (see also [1]).
We will also make use of the alternative version of the Hutchinson operator.

Definition 2.4.
Let F be an IFS on a metric space X. The map F F : 2 X → 2 X defined by: will be called the weak Hutchinson operator of F. Remark 2.5. According to Remark 2.2 (1), if F is finite, then F is compact and the Hutchinson operator coincides with the weak Hutchinson operator on the hyperspace K(X). This is also the case for certain infinite IFSs. For example, if the parameter space I is a compact topological space and F = {f i : i ∈ I} consisting of contractive maps is such that the map X ×I (x, i) → f i (x) ∈ X is continuous, then for any nonempty and compact K, the map i → f i (K) is continuous and hence i∈I f i (K) itself is also compact. Such a version of infinite IFSs were considered for example in [17].
Clearly, in the case when the IFS F is finite, . The next lemma shows some relationships between the Hutchinson operator and the weak Hutchinson operator. It is a folklore, we give the short proof for the sake of completeness. Lemma 2.6. In the above framework, for every n ∈ N and A ⊂ X, Proof. We just have to prove the equality cl F  By continuity of maps from F and standard properties of the closure operator, we have Now assume that the assertion holds for some n ∈ N. Then by the assumption and the above calculations, and the result follows.
We end this section with a simple lemma, which gives a description of the n-th iteration of the weak Hutchinson operator. We skip its proof as it follows directly from definition of the Hutchinson operator.
be IFSs on the same metric space X. Then for every A ⊂ X and a natural number n, the n-th iterate of F F∪G of a set A is given by: where we agree that k 1 can equal 0 (and then f i 1 1 • · · · • f i 1 k 1 disappears), or s p can equal 0 (and then g j p 1 • · · · • g j p sp disappears).

Hutchinson Attractors of Iterated Function Systems
The classical Hutchinson result [2,11] states that: Theorem 2.8. Assume that F is an IFS on a complete metric space X consisting of finitely many Banach contractions (that is, the Lipschitz constant Lip(f ) < 1 for f ∈ F). Then there exists the unique set Moreover, for every D ∈ C b (X), the sequence of iterates of the Hutchinson operator F As we will recall in the following, the above result can be strengthen in various ways-we can weaken the contractive assumptions and allow F to be infinite. On the other hand, we may loose compactness of A F . In order to make siutable formulation, we should give some further notations.
In that case, the map ϕ will be called a witness for f .
Note that a witness function ϕ for f necessarily satisfies ϕ(t) < t for t > 0, so each Matkowski contraction is nonexpansive map. On the other hand, if the Lipschitz constant Lip(f ) < 1, then f is a Matkowski contraction-simply take the function ϕ(t) := Lip(f )t as a witness.
As proved by Matkowski [19], each Matkowski contraction on a complete metric space satisfies the thesis of the Banach fixed point theorem. In fact, this result is one of the strongest generalizations of the Banach theorem. For example, Rakotch contractions or Browder contractions are Matkowski contractions. For a discussion on various types of contractiveness from the perspective of the fixed point theory, see for example [13]. Definition 2.10. We say that an IFS F is a Matkowski contractive, if F is bounded and all maps from F are Matkowski contractions with the same witness. Now we give the promised extension of Theorem 2.8. It is rather a folklore. However, in the literature we could only find its more specialized versions (for example, restricted to countable IFSs), hence we give the sketch of the proof. Theorem 2.11. Assume that F is a Matkowski contractive IFS on a complete metric space X. The following conditions hold.
(1) There exists the unique set A F ∈ C b (X) such that Moreover, for every D ∈ C b (X), the sequence of iterates of the Hutchinson Proof. In a standard way (see a.e., [10, Proposition 1]) we can show that for every f ∈ F and D, Hence F F is a Matkowski contraction on the hyperspace C b (X) w.r.t. the Hausdorff-Pompeiu metric and the existence and uniquness of A F follows from the mentioned Matkowski fixed point theorem. Now if we additionally assume that F is compact, then we can use the Matkowski theorem for the restriction F F |K(X) : K(X) → K(X) and justify that the (necessarily unique) set A F is compact.
Definition 2.12. The set A F which satisfies the assertion of item (1) of the above theorem will be called the Hutchinson attractor of F. Remark 2.13. As proved in [12], if in Theorem 2.11 we additionally assume that the witness ϕ satisfies the following condition lim sup , then there exists a nonempty and bounded setÃ such that F F (Ã) =Ã. Clearly, the attractor of F is then A F = cl(Ã). On the other hand, in [9] there is given a sufficient condition for the attractor

Remark 2.14. Observe that if A F is the Hutchinson attractor of an IFS F, then for every closed and bounded set D ⊂ X, there exists closed and bounded set
Note that the set E can be considered as the attractor of F with condensation D (or inhomogeneous attractor); see, i.e., [2]. Indeed, it is easy to see that Remark 2.15. Let us commend that the class of Hutchinson attractors is very large. In fact, every closed and bounded subset of a metric space A ⊂ X is the Hutchinson attractor of the IFS F = {f y : y ∈ A}, where f y (x) := y for x ∈ X. Moreover, if A is additionally separable, then in a similar way we can show that A is the Hutchinson attractor of some countable IFS. On the other hand, in the main results of the paper, we will prove that considered sets are Hutchinson attractors of certain IFSs given a priori.
The following remark explains some nuances in the definition of Matkowski contractiveness and Theorem 2.11. Remark 2.16. Assume that F is an IFS on a complete metric space X.

If F is finite and consists of Matkowski contractions, then F is Matkowski
contractive. Indeed, it is bounded and its common witness is the function max{ϕ f : f ∈ F}, where ϕ f is a witness for f ; cf. [10].
2 ), but F is not bounded. Observe that here each F-invariant set is necessarily unbounded, so F does not have any Hutchinson attractor.

Semiattractors of Iterated Function Systems
We first give a definition of Kuratowski limits (see, e.g., [4,14] for detailed discussion). Definition 2.17. Let (S n ) be a sequence of subsets of a metric space X. The upper Kuratowski limit of (S n ) is defined by so that x n ∈ S n for infinitely many n ∈ N} and the lower Kuratowski limit of (S n ) is defined by Li(S n ) := {y ∈ X : y = lim n→∞ x n for some sequence (x n ) so that x n ∈ S n for all n ∈ N} If Ls(S n ) = Li(S n ), then its common value, denoted by Lt(S n ), is called the Kuratowski limit of (S n ). Now we recall the notion of semiattractor of an IFS. We refer the reader to [14] or [16] for details.
is nonempty, then we call it as the semiattractor of F, and will denote by A * F . The following result lists basic properties of semiattractors, that will be of importance later on: In particular, Let us end this section with a result on a relationship between the semiattractor of an infinite, countable IFS F and its finite "sub-IFSs" F ⊂ F (see [8] or [20] for further discussion).
The above result shows that the semiattractors (or attractors) of countable infinite IFSs can be approximated by the semiattractors (or attractors) of their finite sub-IFSs. Let us note, however, that there exist infinite IFSs with attractors such that their finite sub-IFSs do not have attractors (see [15,Example 5]).

Chaos Game and Disjunctive Sequences
The chaos game algorithm is a classical and simple algorithm for getting images of Hutchinson attractors of contractive IFSs. In classical setting, when F = {f i : i ∈ I} is a finite IFS, we pick any point x 0 ∈ X, then we choose randomly a map f i1 from the IFS F = {f i : i ∈ I} and define x 1 := f i1 (x 0 ). Then we choose randomly a map f i2 and define x 2 := f i2 (x 1 ), and so on. Then, (with a  [18]) is that with the probability one, we choose a sequence i = (i 1 , i 2 , . . .), such that each finite sequence of elements from I appears in this sequence infinitely many times.
Hence if x 0 ∈ A F (we can make this assumptions for simplicity), then in each set of the form we will find a point from our sequence (x n ) (even infinitely many such points). This idea leads to deterministic version of the chaos game algorithm-we choose a sequence i = (i 1 , i 2 , . . .) with the above property and generate the sequence (x n ) according to the formula x n := f in (x n−1 ). Such an approach can be adjusted to any countable IFS, and was studied in [15] (see also [5] for more general cases).
Let us make some precise definitions: -The sequence i = (i 1 , i 2 , . . .) of elements of I with the property that each finite sequence appears in i infinitely many times will be called disjunctive.
, where x 0 ∈ X is any point, then i will be called as the driver of (x n ). -We will say that ( where h is the Hausdorff-Pompeiu metric. Remark 2.23. Since any sequence i has countably many values, we see that disjunctive sequences exist only if I is countable. Thus in the given later Theorem 4.1 item (D) we deal just with countable IFSs.
The mentioned explanation of the chaos game algorithm can be reformulated in the following form: the Hutchinson attractor A F is recovered by any sequence (x n ) with disjunctive driver.
Remark 2.24. Let us remark that in [16], a bit different definition of recoverness is considered. Namely, it is stated that the sequence (x n ) recovers the semiattractor A * F , if the set ω((x n )) of limit points of the sequence (x n ) coincides with A * F . Clearly, (2) for A * F implies that ω((x n )) = A * F . On the other hand, in most cases the converse also holds (for example, if one of maps from an IFS is Matkowski contraction and the driver is disjunctive). For a discussion of the concept of recovering we refer to [6].

Description of Semiattractors of Enriched IFSs
Let us introduce the following notation: for an IFS G = {g j : j ∈ J}, let where Id X is the identity function on X. In orther words, M(G) is the monoid generated by G.
The following result shows that the semiattractor of F ∪ G can be described in several ways.
F∪G is the semiattractor of N and in particular: Then A * F∪G is the semiattractor of both M 1 and M 2 , and in particular:   [16]. The theses of the above Theorem 3.1 are partial extensions of main results from that paper. In the next section we will formulate results under more restrictive assumptions, which will give full generalizations of those from [16]. Proof.
The inclusion "⊃" follows easily from the fact that F ∪ G ⊂ N . Now to see the opposite inclusion, take any ingredient in the union n∈N F  (1), that is, a set of the form . Then consider two cases: (1) for sufficiently large p (indeed, we just have to rewrite each h i k as a composition of maps from F and G). Thus All in all, we proved that which gives us the second inclusion in (4). Now we prove item (2). In view of Theorem 2.19, we only have to prove that Using item (1) for N := M(G) ∪ F, and noting that for every A ⊂ X, we see that for every k ∈ N,

N (A)
where in the last equality we used (4). This gives the last inclusion "⊂" in (5). The second inclusion follows from M 2 ⊂ M 1 . Now we show the first one. Take any ingredient from the union n∈N F (n) F∪G (A * F ) of the form (1). Consider two cases: Case 1. The considered ingredient ends with some f from F. Then we can rewrite it in the following way: that is, we replace each f i either by (Id X •f i ), or, if it has elements from G on the left, by (g • f i ), where g is an appropriate composition from maps from G.
In particular, this ingredient is a subset of some F (m) M2 (A * F ). Case 2. The ingredient ends with some map g from G. Then using similar replacement as in the Case 1 and using the fact that A * F = F F (A * F ), we can proceed in the following way: All in all, we proved that which gives us the first inclusion in (5 Again, using item (1) for N := F ∪ M(G), we see that for any k ∈ N, so using Lemma 2.6, we get the inclusion "⊂" in (6). We will show the opposite one. Take any ingredient from the union n∈N F (n) F∪G (A * F ) of the form (1). Consider two cases: Case 1. There appears some map f from F in the composition in (1). Then we can rewrite it in the following way: that is, we add Id X at the beginning (if it is needed) and replace each f i either by (f i • Id X ), or, if it has elements from G on the right, by (f i • g), where g is an appropriate composition from maps from G. In particular, this ingredient is a subset of some The ingredient is of the form g i1 • · · · • g in (A * F ). Then we also have All in all, we proved that which gives us the second inclusion in (6)  Now we move to item (4). We first show (4i). Since A * F∪G is the semiattractor of F ∪ M(G) (see item (1)), we have g(A * F∪G ) ⊂ A * F∪G for every g ∈ M(G). Hence, taking inverse function g −1 , we also have A * F∪G = g(g −1 (A * F∪G )) ⊂ g(A * F∪G ). Now we prove item (4ii). Since R ⊂ M 1 from item (2), we see that It remains to show the opposite inclusion. Take any ingredient of the union . We can rewrite it, and using item (1), proceed as follows: Hence using item (2), we have and this gives us the remaining inclusion in (3). We and this section with the question concerning item (4) of Theorem 3.1.

Problem 3.4. In the framework of item (4) of Theorem 3.1, is the set A * F∪G the semiattractor of the IFS R?
Note that in the next section we give sufficient conditions for which this is the case.

Enriching Matkowski Contractive IFSs with Nonexpansive Mappings
In this section we prove that under additional contractive assumptions on IFSs F and G, we can extend the assertions of Theorem 3.1 by observing that considered IFSs M 1 , M 2 , H and R are actually Matkowski contractive.
Then: (Ci) H is Matkowski contractive (with the same witness as for F). Proof. Item (A) follows from (B) (and (C)) Now we prove (B). We first show that M 1 is a bounded IFS. Take any closed and bounded set D ⊂ X. We have and the latter set is bounded as F ∪ M(G) is bounded. Hence M 1 is bounded. Now take f ∈ F and g, h ∈ M(G), and let ϕ be a common witness for maps from F. Then for every x, y ∈ X, we have (g(x), g(y))) ≤ ϕ(d(x, y)) To see (D), first fix an enumeration F ∪G = {h i : i ∈ I} and let ϕ be a witness function of maps from F. Then we have (see a.e., [15]) where t n is the number of maps from F used in the driver of (x n ) up to n-th position. Finally, choose any ε > 0. Since (t n ) → ∞ when n → 0 (as the driver is disjunctive), we have that d(x n , A * F∪G ) → 0 and hence we can find k 1 ∈ N such that for every n ≥ k 1 , we have Fix any k ≥ k 1 . Now choose a closed and bounded set D such that x 0 ∈ D, A * F∪G ⊂ D and F M2 (D) ⊂ D (this can be done since M 2 is Matkowski contractive, see Remark 2.14). Then the whole sequence (x n ) ⊂ D. Take any z ∈ A * F∪G and find k 0 ∈ N so that By Theorem 3.1 item (2) and Theorem 2.19 item 2., and the fact that the sequence (F (n) M2 (A F )) is nondecreasing, we can find j ≥ k 0 and z 1 ∈ F (j) Then by Lemma 2.6, we can find z ∈ F (j) we can find maps f 1 , . . . , f j ∈ F and g 1 , . . . , g j ∈ M(G) such that Now, as each g i is a composition of maps from G, we can go back to enumeration of F ∪ G = {h i : i ∈ I}, and find i 1 , . . . , i p such that Since the driver of (x n ) is disjunctive, there is an n 0 , which can be assumed to be greater than earlier taken k, such that As the attractor A F ⊂ A * F∪G ⊂ D, we see that also the earlier chosen point z All in all, By (7) and (8), we have that h({x n : n ≥ k}, A * F∪G ) ≤ ε and the result follows. Finally, observe that (Ei) is clear, and (Eii) follows from (Ei) and the fact that the Hutchinson attractor of a compact IFS is compact (Theorem 2.11 item (2)).
As a simple corollary of items (A) and (E) of the above theorem, we get:  (3), we see that in Theorem 4.1, the assumption that M(G) is bounded means exactly that there exists z 0 ∈ X such that the orbit {g(z 0 ) : g ∈ M(G)} is bounded. In the next result we show that boundedness of M(G) is essential.   In the next theorem we study the case when the monoid of M(G) is actually a group. Note that this assumption together with nonexpansiveness of elements of G, force that M(G) is group of isometries, so we will formulate it in such a way. Theorem 4.6. Assume that F, G are IFSs on a complete metric space such that: • F is Matkowski contractive; • the monoid M(G) is bounded and is a group of isometries. Let A * F∪G be the semiattractor of F ∪ G. The the following conditions hold: Then (1i) the IFS R is Matkowski contractive (with the same witness as for F); (1ii) the Hutchinson attractor of R coincides with A * F∪G ; that is, Then the following conditions are equivalent:  (2). The equivalence of (i) and (ii) follows from the fact that A * F∪G is closed and bounded. The equivalence of (iv) and (v) follows from definition of M(G). The implication (iii) ⇒ (i) follows from Theorem 4.1 item (C). The implication (iii) ⇒ (v) follows from item (1). Now to see (iv) ⇒ (iii), observe that Hence by Lemma 2.6 and standard properties of closures, we have (in fact, we could also use the fact that A F is closed and bounded F ∪ Ginvariant set). Finally, we prove the remaining implication (i) ⇒ (iii). LetÃ = g∈M(G) g(A H ). Then by Lemma 2.6 and Theorem 3.1 (3), we have Hence A * F∪G is the attractor of F as it is closed and bounded.   (2) of Theorem 4.6 holds, then the semiattractor A * F∪G is compact. In the next section we will give a relatively simple example in which the semiattractor A * F∪G is not compact. We now show that main results from [16] can be obtained as corollaries of our theorems. Below we let g (0) to be the identity map.
Finally, we present an example which shows that the assumption on nonexpansiveness of maps from G is important in the above discussion:

Example of Noncompact Bounded Semiattractor A * F∪G
We say that a selfmap f : X → X of a metric space X is rotative (see [7]), if there exist n ∈ N and α < n such that for every x ∈ X, In such case we call f as an (α, n)-rotative. Let X := ∞ (C), that is, X is the Banach space of all bounded complex sequences endowed with the supremum norm || · ||. Set

Lower Transition Attractors as Semiattractors of Enriched IFSs
We start with giving some further denotations from [21]. Throughout this section, let It is clear that from the perspecive of invesigations of lower transitions attractors, we can assume that t 0 = 1 (just consider the scaled maps t 0 f 1 , ..., t 0 f N , t 0 g). In such a case, it is easy to see that the maps g and g 1 are isometries. The following result shows that the lower transition attractor is the semiattractor of F 1 ∪ {h} for certain appropriate constant map h and, in consequence, the Hutchinson attractor of certain IFSs. 1 (a * ) = a * ; recall Remark 2.2), hence compact (as R d is a proper space). Hence, using Theorem 4.1 items (A) and (E) for {f (1,1) , . . . , f (N,1) , h} and {g 1 }, we see that the semiattractor A * F1∪{h} is compact. Moreover, using Theorem 2.19 for IFSs {h} and F 1 , we see that the semiattractor of F 1 ∪ {h} equals Now since a * is a fixed point of g 1 , we see that a * ∈ F Remark 7.3. It can be wieved that the above presented proof can be exetended into wider setting. In forthcoming paper we are going to invesigate the existence of lower transition attractors (as well as upper transition attractors) from [21] for more general families F t .